We construct a 1-parameter family of noncommutative Hopf fibrations
via 2-cocyle deformations. We refer the reader to [1]} for a
general cocycle deformation framework and also to [2]}, [3]}, [4]}
for the more specific case of toric deformations, which is sufficient for our present work.
Let \(K = \mathcal {O}\big (\mathbb {T}^2\big )\)  denote the \(\ast \) -Hopf algebra of
functions on the algebraic 2-torus. As a vector space,
\(K\)  is spanned by the basis
\(\big \lbrace t_{\mathbf {m}}\colon \mathbf {m} = (m_1,m_2) \in \mathbb {Z}^2\big \rbrace  ,\) 
